The 2.8.2 release of DragonFly BSD is now available, featuring significant advances in multi-processor performance based on DragonFly's signature soft token locks. It also includes many feature advancements including: pf from OpenBSD 4.2, the Wifi stack from FreeBSD and DataMapper from NetBSD (with significant enhancements). This release also marks the return of the GUI image. See the release notes for full details.

Is there binary compatibility? Between DragonflyBSD and FreeBSD for instance? Or some sort of "acceptance" in the sense of one system accepting packages built for an ancestor?

No, there's no binary compat between the BSDs. They are all separate OSes, with separate ABIs/APIs/etc. Just like you can't (normally) run a RHEL 4.x binary on a Debian 5.x system, you can't run an OpenBSD binary on FreeBSD.

I guess these four flavors are desktop OSes?

You guess wrong. While they can be used as desktop OSes, and run the same GUI stack as Linux (Xorg, GNOME/KDE, XFce, etc), they are developed primarily as server OSes.

Is the user base significant? At what level should BSD as a whole be placed? The same as Linux?

Considering all but DragonflyBSD are older than most (if not all) Linux distributions, I'd place them on the same level. lol However, if you just go by market share and mind share, they're one rung lower than Linux, but way ahead of all the rest of the alternative OSes.